Lambert Conformal Conic to Geographic Transformation Formulae

This page explains how to convert Lambert Conformal Conic projection coordinates ( N , E ) to their geographic equivalents and vice versa.

Projection parameters

The equations on this page use the following parameters which are specific to the particular projection that is being converted to or from. The correct values for different New Zealand projections can be found in the projections section.

aSemi-major axis of reference ellipsoid
fEllipsoidal flattening
symbol-phi-1Latitude of first standard parallel
symbol-phi-2Latitude of second standard parallel
symbol-phi-0Origin latitude
symbol-lambda-0Origin longitude
symbol-n-0False Northing
symbol-e-0False Easting
symbol-phiLatitude of computation point
λLongitude of computation point
NNorthing of computation point
EEasting of computation point

The following equations are divided into three sections:

Projection constants

Several additional parameters need to be computed before transformations can be undertaken (symbol-little-e,symbol-little-n,symbol-f,symbol-rho-0). These parameters are constant for a projection.

equation-little-e

equation--little-n

equation-f

equation-rho

where:

equation-m

equation-t

symbol-m1 and symbol-m2are obtained by evaluating symbol-musing symbol-phi-1 and symbol-phi-2,

symbol-t0symbol-t1 and symbol-t2are obtained by evaluating symbol-little-t using symbol-phi-0 symbol-phi-1 and symbol-phi-2,

 symbol-rho-0 is obtained by evaluating symbol-rhousing symbol-t0

Geographic to Lambert conformal projection

The conversion of geographic coordinates (symbol-phisymbol-lambda) to projection coordinates ( N , E ) is achieved in several steps. First, determine symbol-little-t and symbol-rho

at the using the latitude of the computation point (symbol-phi) and the formulas above. Then evaluate symbol-theta at the longitude of the computation point (symbol-lambda) using:

equation-y

The projection northing ( N ) of the computation point is computed using:

equation-n

Finally the projection easting ( E ) of the computation point is computed using:

equation-e

Lambert conformal projection to geographic

The conversion of Lambert projection coordinates ( N , E ) to geographic coordinates ( f , l ) is achieved in several steps. First, determine symbol-n-1symbol-e-1 ,

symbol-rho-1symbol-t-sup-1 and symbol-theta-1 using the following formulas:

equation-n-1

equation-e-1

equation-rho-1-new

equation-t-1

equation-y-1

The latitude of the computation point needs to be computed iteratively. The first approximation is obtained from:

equation-phi-first-approx

This initial estimate of symbol-phi

is then substituted into:

equation-phi

This value of symbol-phi should be re-substituted into the above formula until successive values do not change. This is typically achieved after three iterations.
The longitude of the computation point (symbol-lambda) is determined using:

equation-lambda